Washington (CNN)It's no secret that Washington has long been divided on new gun control laws. But Americans themselves might not be so divided (at least, on a few things) after all.

There are many areas of gun policy that split the country, but majorities in both parties do come together to support several key gun control measures, according to a survey this spring from the Pew Research Center.

Broad majorities of more than eight in 10 Republicans and Democrats (and independents who lean toward each party) support blocking people who are mentally ill and people on federal no-fly or watch lists from buying guns.

Majorities also favor background checks for private and gun show sales, though the number of Republicans and GOP-leaning independents falls slightly to three in four on this potential policy.

And, though only by a slim margin among Republicans, majorities of both parties even back an assault weapons ban and creating a new federal database to track gun sales. (Support among Democrats and Democratic-leaning independents for these priorities remains around eight in 10, creating a broad gap between the two parties' support.)

Read More

Still, polls also show that, when asked a general question about whether to create stricter gun laws, most Republicans oppose that direction, while most Democrats support it. Despite multiple high-profile mass shootings over the last decade, Congress has not passed recent gun control legislation.

Democrats are much more likely to see gun violence as a very big problem, according to Pew's findings.

Questions on increasing the availability of guns -- like allowing concealed carry in more places and allowing teachers in schools to carry -- received the support of majorities of Republicans, but only about a quarter of Democrats.

Two of the five deadliest mass shootings in modern US history have occurred in the last two months. A staggering 58 people were killed and more than 500 were hurt when a shooter opened fire on a Las Vegas concert and at least 26 people were killed in a Texas church on Sunday morning.

President Donald Trump, on a foreign trip to Asia when the Texas shooting occurred, said mental health care should be the priority in the wake of the attack.

"I think that mental health is your problem here," he said at a news conference. "This isn't a guns situation. I mean, we could go into it, but it's a little bit soon to go into it. ... But this is a mental health problem at the highest level."

Texas Gov. Greg Abbott told CNN on Monday morning that the shooter "sought to get a license to carry a gun in the state of Texas, but the state denied him the ability to get one," stressing that it was still early in the investigation.

This Pew Research Center survey was conducted from March 13-27 and April 4-18, 2017 among 3,930 adults. The margin of sampling error is ±2.8 percentage points for the full sample; it is larger for subgroups like leaned Republicans and leaned Democrats.

UPDATE: This piece has been updated to reflect Sunday's shooting in Sutherland Springs, Texas.